Lecture 8 - Model Identification
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•
•
•
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What is system identification?
Direct pulse response identification
Linear regression
Regularization
Parametric model ID, nonlinear LS
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What is System Identification?
Experiment
Plant
Data
Identification
Model
• White-box identification
– estimate parameters of a physical model from data
– Example: aircraft flight model
• Gray-box identification
Rarely used in
real-life control
– given generic model structure estimate parameters from data
– Example: neural network model of an engine
• Black-box identification
– determine model structure and estimate parameters from data
– Example: security pricing models for stock market
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Industrial Use of System ID
• Process control - most developed ID approaches
– all plants and processes are different
– need to do identification, cannot spend too much time on each
– industrial identification tools
• Aerospace
– white-box identification, specially designed programs of tests
• Automotive
– white-box, significant effort on model development and calibration
• Disk drives
– used to do thorough identification, shorter cycle time
• Embedded systems
– simplified models, short cycle time
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Impulse response identification
• Simplest approach: apply control impulse and collect the
data
IMP ULS E RES P ONS E
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
TIME
• Difficult to apply a short impulse big enough such that the
response is much larger than the noise
NOIS Y IMP ULS E RES P ONS E
1
0.5
0
-0.5
0
1
• Can be used for building simplified
control design models from complex sims
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3
4
5
6
TIME
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7
Step response identification
• Step (bump) control input and collect the data
– used in process control
S TEP RES P ONS E OF P AP ER WEIGHT
1.5
1
Actuator bumped
0.5
0
0
200
400
600
TIME (S EC)
800
1000
• Impulse estimate still noisy: impulse(t) = step(t)-step(t-1)
IMP ULS E RES P ONS E OF P AP ER WEIGHT
0.3
0.2
0.1
0
0
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TIME (S EC)
500
600
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Noise reduction
Noise can be reduced by statistical averaging:
• Collect data for mutiple steps and do more averaging to
estimate the step/pulse response
• Use a parametric model of the system and estimate a few
model parameters describing the response: dead time, rise
time, gain
• Do both in a sequence
– done in real process control ID packages
• Pre-filter data
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Linear regression
• Mathematical aside
– linear regression is one of the main System ID tools
Data
Regression weights
Regressor
N
y (t ) = ∑θ jϕ j (t ) + e(t )
Error
y = Φθ + e
j =1
 y (1) 
 ϕ1 (1) K ϕ K (1) 
 θ1 
 e(1) 
O
M , θ =  M , e =  M 
y =  M , Φ =  M




 


 y ( N )
ϕ1 ( N ) K ϕ K ( N )
θ K 
e( N )
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Linear regression
• Makes sense only when matrix Φ is
tall, N > K, more data available than
the number of unknown parameters.
– Statistical averaging
• Least square solution:
||e||2
→ min
y = Φθ + e
θˆ = (Φ T Φ ) Φ T y
−1
– Matlab pinv or left matrix division \
• Correlation interpretation:
N
 N 2

K ∑ ϕ K (t )ϕ1 (t )
 ∑ϕ 1 (t )
t =1
1  t =1

,
M
O
M
R= 

N
N N
2
∑ ϕ1 (t )ϕ K (t ) K

ϕ
∑
K (t )
 t =1

t =1
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θˆ = R −1c
 N

 ∑ ϕ1 ( t ) y ( t ) 
1  t =1

M
c= 

N N
 ∑ ϕ K (t ) y (t )
 t =1

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Example: linear first-order model
y (t ) = ay (t − 1) + gu (t − 1) + e(t )
• Linear regression representation
ϕ1 (t ) = y (t − 1)
a 
θ = 
ϕ 2 (t ) = u(t − 1)
g
−1 T
T
ˆ
θ = (Φ Φ ) Φ y
• This approach is considered in most of the technical
literature on identification
Lennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999
• Matlab Identification Toolbox
– Industrial use in aerospace mostly
– Not really used much in industrial process control
• Main issue:
– small error in a might mean large change in response
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Regularization
• Linear regression, where Φ T Φ is ill-conditioned
• Instead of ||e||2 → min solve a regularized problem
2
2
e + r θ → min
y = Φθ + e
r is a small regularization parameter
• Regularized solution
−1 T
T
ˆ
θ = (Φ Φ + rI ) Φ y
• Cut off the singular values of Φ that are smaller than r
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Regularization
• Analysis through SVD (singular value decomposition)
Φ = USV T ;
V ∈ R n ,n ;U ∈ R m ,m ; S = diag{s j }nj =1
• Regularized solution
n




s
−
1


θˆ = (Φ T Φ + rI ) Φ T y = V diag  2 j  U T y
 s j + r  j =1 



• Cut off the singular values of Φ that are smaller than r
REGULARIZED INVERS E
2
1.5
s
s 2 + 0.1
1
0.5
0
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s
4
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Linear regression for FIR model
P RBS EXCITATION S IGNAL
• Identifying impulse response by 1
0.5
applying multiple steps
0
-0.5
• PRBS excitation signal
-1
• FIR (impulse response) model 0
K
y ( t ) = ∑ h ( k )u ( t − k ) + e( t )
k =1
10
20
30
40
PRBS =
Pseudo-Random Binary Sequence,
see IDINPUT in Matlab
• Linear regression representation
ϕ1 (t ) = u(t − 1)
M
ϕ K (t ) = u (t − K )
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 h (1) 
θ = M 


h ( K )
Control Engineering
−1 T
T
ˆ
θ = (Φ Φ + rI ) Φ y
8-12
50
Example: FIR model ID
P RBS e xcita tion
• PRBS excitation
input
1
0.5
0
-0.5
-1
0
200
400
600
800
1000
800
1000
S YS TEM RES P ONS E
• Simulated system 1
output: 4000
0.5
samples, random 0
-0.5
noise of the
-1
amplitude 0.5
0
200
400
600
TIME
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Example: FIR model ID
FIR es tima te
• Linear regression 0.2
estimate of the FIR0.15
0.1
model
0.05
0
-0.05
0
1
2
3
4
5
6
7
5
6
7
Impuls e Re s pons e
• Impulse response
for the simulated
system:
0.2
0.15
0.1
0.05
0
-0.05
T=tf([1 .5],[1 1.1 1]);
0
P=c2d(T,0.25);
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Time (s ec)
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Nonlinear parametric model ID
• Prediction model depending on
the unknown parameter vector θ
u (t ) → MODEL(θ ) → yˆ (t | θ )
Loss Index V
V = ∑ y (t ) − yˆ (t | θ )
• Loss index
J = ∑ y (t ) − yˆ (t | θ )
θ
Optimizer
2
2
• Iterative numerical optimization.
Computation of V as a subroutine
y (t )
u (t )
Model including the
parameters θ
sim
Lennart Ljung, “Identification for Control: Simple Process Models,”
IEEE Conf. on Decision and Control, Las Vegas, NV, 2002
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Parametric ID of step response
τ
• First order process with deadtime
• Most common industrial process model
• Response to a control step applied at tB
 g (1 − e( t −tB −TD ) /τ ), for t > tB − TD
y (t | θ ) = γ + 
0,
for t ≤ tB − TD

γ 
g
θ = 
τ 
 
TD 
g
TD
Example:
Paper
machine
process
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Gain estimation
• For given τ , TD , the modeled step response can be
presented in the form
y (t | θ ) = γ + g ⋅ y1 (t | τ , TD )
• This is a linear regression
2
w1 = g
k =1
w2 = γ
y (t | θ ) = ∑ wkϕ k (t )
ϕ1 (t ) = y1 (t | τ , TD )
ϕ 2 (t ) = 1
• Parameter estimate and prediction for given τ , TD
wˆ (τ , TD ) = (Φ Φ ) Φ T y
T
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yˆ (t | τ , TD ) = γˆ + gˆ ⋅ y1 (t | τ , TD )
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Rise time/dead time estimation
• For given τ , TD , the loss index is
N
V = ∑ y (t ) − yˆ (t | τ , TD )
2
t =1
• Grid τ , TD and find the minimum of V = V (τ , TD )
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Examples: Step response ID
• Identification results for real industrial process data
• This algorithm works in an industrial tool used in 500+
industrial plants, many processes each
P roce s s pa ra me te rs : Ga in = 0.134; Tde l = 0.00; Tris e = 119.8969
1
1.6
1.4
Nonlinear
Regression ID
0.8
0.6
1.2
1
0.4
0.8
Linear
Regression ID
of the first-order
model
0.2
0
-0.2
-0.4
0
10
20
30
40
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70
0.6
Nonlinear
Regression ID
0.4
0.2
0
80
-0.2
0
100
200
300
400
500
600
700
time in s e c.; MD re s pons e - s olid; e s tima te d re s pons e - da s he d
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Linear filtering
• A trick that helps: pre-filter data
• Consider data model
y = h *u + e
• L is a linear filtering operator, usually LPF
Ly = L( h * u ) + {
Le
{
e
yf
f
L( h * u ) = ( Lh ) * u = h * ( Lu )
• Can estimate h from filtered y and filtered u
• Or can estimate filtered h from filtered y and ‘raw’ u
• Pre-filter bandwidth will limit the estimation bandwidth
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Multivariable ID
• Apply SISO ID to various input/output pairs
• Need n tests - excite each input in turn
• Step/pulse response identification is a key part of the
industrial Multivariable Predictive Control packages.
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